Integrand size = 37, antiderivative size = 78 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac {2 e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{3 d} \]
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Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {706, 703, 227} \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {2 e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{3 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{3 d} \]
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Rule 227
Rule 703
Rule 706
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac {1}{3} e^2 \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {2 e \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac {(2 e) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{3 d} \\ & = -\frac {2 e \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac {2 e^{3/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.69 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e \sqrt {e (c+d x)} \left (\sqrt {1-(c+d x)^2}-\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(64)=128\).
Time = 2.76 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.03
method | result | size |
default | \(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e \left (-2 d^{3} x^{3}-6 c \,d^{2} x^{2}+\sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-6 c^{2} d x -2 c^{3}+2 d x +2 c \right )}{3 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\) | \(158\) |
elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \left (-\frac {2 e \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{3 d}+\frac {2 \left (c^{2} e^{2}+\frac {2 e \left (-\frac {3}{2} d e \,c^{2}+\frac {1}{2} d e \right )}{3 d}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\left (d x +c \right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e}\) | \(359\) |
risch | \(\frac {2 \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{2}}{3 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {2 \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{2}}{3 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(395\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e} d^{2} e + \sqrt {-d^{3} e} e {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{3 \, d^{3}} \]
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\[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
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\[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {3}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]
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\[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {3}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^{3/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
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